# Why doesn't .9999~ = 1? **URL:** \ **Category:** Factual Questions\ **Created:** [June 1, 2003, 11:12pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019 "2003-06-01T23:12:58Z")\ **Posts on this page:** 20\ **Page:** 16
**Author:** ![BrotherCadfael](https://avatars.discourse-cdn.com/v4/letter/b/977dab/32.png) [@BrotherCadfael](https://boards.straightdope.com/u/BrotherCadfael)\ **Post date:** [June 4, 2003, 8:05pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/301 "2003-06-04T20:05:58Z")
FranticMad – The sensation you are experiencing is called “algorasm”. It’s why pure mathematics is so much fun. ---
**Author:** ![Flymaster](https://avatars.discourse-cdn.com/v4/letter/f/c4cdca/32.png) [@Flymaster](https://boards.straightdope.com/u/Flymaster)\ **Post date:** [June 4, 2003, 8:19pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/302 "2003-06-04T20:19:28Z")
> [@](#): > > \*Originally posted by Pleonast \* > \*\*It is easy to prove that any decimal representation (with a finite number of digits left of the decimal, and an uncountable number of digits to the right) always converges (let’s call this the Decimal Convergence Theorem). I agree with the conclusion here, but not the statement. Clearly, it’s fairly easy to prove that a decimal converges. However, there are NOT an UNcountable number of digits to the right. It is, in fact, a very countable infinite sequence. Unless I’m missing something. Am I? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 4, 2003, 8:23pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/303 "2003-06-04T20:23:24Z")
No, it’s countable. **Pleonast** may have been using “uncountable” in the sense of “not countable in finite time”, which is bad but understandable. ---
**Author:** ![Flymaster](https://avatars.discourse-cdn.com/v4/letter/f/c4cdca/32.png) [@Flymaster](https://boards.straightdope.com/u/Flymaster)\ **Post date:** [June 4, 2003, 8:29pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/304 "2003-06-04T20:29:03Z")
That’s what I assumed, but in an argument as full of mathematical pedantry as this one, I figured that shouldn’t go uncaught. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [June 4, 2003, 8:43pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/305 "2003-06-04T20:43:48Z")
I always hated that term. Why on earth would something with cardinality Aleph-Null be called “countable”? Couldn’t they come up with a Latin word for it instead? ---
**Author:** ![Flymaster](https://avatars.discourse-cdn.com/v4/letter/f/c4cdca/32.png) [@Flymaster](https://boards.straightdope.com/u/Flymaster)\ **Post date:** [June 4, 2003, 8:52pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/306 "2003-06-04T20:52:38Z")
Because, given infinite time, every item in the sequence/set can be counted. If it’s uncountable, then infinite time isn’t enough to count every item in the set. ---
**Author:** ![ragerdude](https://avatars.discourse-cdn.com/v4/letter/r/d07c76/32.png) [@ragerdude](https://boards.straightdope.com/u/ragerdude)\ **Post date:** [June 4, 2003, 8:56pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/307 "2003-06-04T20:56:33Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*…After all, it’s intuitive that .9… != 1, but that’s not true. … \*\* I guess everyone’s intuition is different. Maybe my “intuition” has been influenced by my education, but I’ve always thought that it’s intuitive that .999… = 1. I can’t comprehend it being anything else. Any other identification makes no sense. And, **Achernar** , it seems as though introducing Latin terms causes mass confusion. 😉 ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [June 4, 2003, 9:22pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/308 "2003-06-04T21:22:02Z")
I also find it intuitive that 0.999… = 1, but **ultrafilter’s** point still stands. Read about the [Banach-Tarski Paradox](http://mathworld.wolfram.com/Banach-TarskiParadox.html) and then say that intuition is a good guide to mathematical truth. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [June 4, 2003, 9:23pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/309 "2003-06-04T21:23:14Z")
> [@](#): > > Pleonast may have been using “uncountable” in the sense of “not countable in finite time” Doh! I meant “uncountable” in the sense of “countable, but my fingers typed the wrong word”! Sorry about that, I really do know the difference. > [@](#): > > I (and the other math types will probably agree with me here) really don’t like relying on intuition. After all, it’s intuitive that .9… != 1, but that’s not true. Let’s be explicit in all of our steps. I agree it dangerous relying on intuition. But how explicit must we be? Even **Phage** (I think?) accepts that 0.999… converges (he just disagrees what it converges to). And does anyone else find the theorem “a==b && c==d =\> a-c == b-d” unintuitive? I do, yet no one has complained about the failure to mention it in step 3 of [**RealityChuck** ’s proof](http://boards.straightdope.com/sdmb/showthread.php?threadid=187902&perpage=50&pagenumber=4#post3510569). Once we begin to leave off the justification for any step in a proof, we’ve opened the door for “stylistic” differences in our choices of what to leave implicit, and what to make explicit. ---
**Author:** ![RufusLeaking](https://avatars.discourse-cdn.com/v4/letter/r/c5a1d2/32.png) [@RufusLeaking](https://boards.straightdope.com/u/RufusLeaking)\ **Post date:** [June 4, 2003, 9:58pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/310 "2003-06-04T21:58:25Z")
1/3 is expressed as .333… because of a limitation of the base 10 number system not because there is anything magic about the number. In the base 9 number system … 1/3 = .3 1/6 = .6 So 1/3 + 1/6 = .3 + .6 = 1.0 The best we can do in base ten is say that 1/3 is the limit that .333… is approaching. So the limit that .999… is approaching just happens to be a number that can be expressed in base 10 as 1.0. ---
**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\ **Post date:** [June 4, 2003, 9:59pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/311 "2003-06-04T21:59:34Z")
Mods, Can I start a book on when this thread will end? (I’ll split with ya!) ---
**Author:** ![Saltire](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saltire/32/553_2.png) [@Saltire](https://boards.straightdope.com/u/Saltire)\ **Post date:** [June 4, 2003, 10:10pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/312 "2003-06-04T22:10:19Z")
> [@](#): > > \*Originally posted by The Great Unwashed \* > \*\*Mods, > > Can I start a book on when this thread will end? > > (I’ll split with ya!) \*\* I’ll put 1/infinity dollars on July 32nd at 24:03 hours! ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [June 4, 2003, 10:13pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/313 "2003-06-04T22:13:00Z")
I’ll take $_i_-1 on June 10th at 10:56 AM ---
**Author:** ![William\_Ashbless](https://avatars.discourse-cdn.com/v4/letter/w/a87d85/32.png) [@William\_Ashbless](https://boards.straightdope.com/u/William_Ashbless)\ **Post date:** [June 4, 2003, 10:16pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/314 "2003-06-04T22:16:35Z")
I’ll take sqrt(-peanut\_butter) on the hrung collapse of Betelgeuse 7. ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [June 4, 2003, 10:17pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/315 "2003-06-04T22:17:05Z")
> [@](#): > > \*Originally posted by Exapno Mapcase \* > \*\*Wow. That’s brilliant. Short, neat, devastating, and utterly obvious once someone smarter than you points it out. > > Now why couldn’t you have come up with that thought 6 pages ago? 🙂 > > P.S. If you’re so freakin’ brilliant, howcum your user name is misspelled? 😕 \*\* typo, didn’t notice until I tried to log in again. I’ve considered changing my name but means losing my post count. ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [June 4, 2003, 10:18pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/316 "2003-06-04T22:18:28Z")
No, the administrators can change your name, and you won’t lose any stats. Email **TubaDiva** , and she’ll take care of it for you. ---
**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\ **Post date:** [June 4, 2003, 10:21pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/317 "2003-06-04T22:21:26Z")
thanx, it’s been bugging me for quite a while 🙂 ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 4, 2003, 10:24pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/318 "2003-06-04T22:24:54Z")
Quoth **Jabba** : > [@](#): > > Read about the Banach-Tarski Paradox and then say that intuition is a good guide to mathematical truth. [hijack, though more on topic than Latin terminology] Does the B-T theorem rely on the Axiom of Choice? Because the pieces the ball is cut into are obviously unmeasureable, and the last time I saw an unmeasureable set constructed (on this board, in fact), the Axiom of Choice was required.[/hijack, though more on topic than Latin terminology] And while I’ll certainly agree that one shouldn’t rely on or trust one’s intuition in mathematics, I wouldn’t go so far as to say that it’s useless. Intuition can be a very good guide for suggesting what problems to attempt, and how to go about attempting them. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [June 4, 2003, 10:30pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/319 "2003-06-04T22:30:02Z")
(i) Yes, it does (ii) I agree, intuition is vital for success in mathematics, but it must be the sort of intuition one develops through years of rigorous training. Man-on-the-Clapham-omnibus intuition is useless. ---
**Author:** ![Nanu](https://avatars.discourse-cdn.com/v4/letter/n/dec6dc/32.png) [@Nanu](https://boards.straightdope.com/u/Nanu)\ **Post date:** [June 4, 2003, 10:54pm UTC](https://boards.straightdope.com/t/why-doesnt-9999-1/179019/320 "2003-06-04T22:54:52Z")
I suppose the answer would be in solving the infinite series of sums: ``` limit when n tends to infinite of: sum(from i=1 to i=n) of 9/10(to i power) ``` I’m not too good with infinite sums but if someone could solve it he would end this dilemma. [Previous page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=15) [Next page](https://boards.straightdope.com/t/why-doesnt-9999-1/179019.md?page=17)